I'm trying to solve the following exercise from an introductory statistics textbook:

Let $X_1, \dots , X_m$ denote a random sample from the exponential density with mean $\theta_1$ and let $Y_1 ,\dots , Y_n$ denote an independent random sample from an exponential density with mean $\theta_2$ .

(a) Find the likelihood ratio criterion for testing $H_0$ : $\theta_1=\theta_2$ versus $H_a$ : $θ_1\neq θ_2$ .

(b) Show that the test in part (a) is equivalent to an exact F test.

I'm fine with part (a), I got simply a test of the form $\frac{X^mY^n}{(X+Y)^{m+n}}\leq k$, where $X=\sum X_i$ and $Y=\sum Y_i$. However, I have no idea how to transform this into an F-test.


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